A Brief History of Geometry

by Allen Glines

November 11, 2009

When you’re flipping through your Geometry textbook, think fondly of cavemen, who discovered obtuse triangles (a triangle where one of the interior angles is larger than 90 degrees) in the Indus Valley, and ancient Babylonia from around 3000 B.C. The earliest recorded beginnings of Geometry can be traced back to those two groups after all. Theories back then consisted of principles concerning lengths, angles, areas, and volumes, which were all derived from experiments and experiments, and developed to meet the particular craft. These theories were so sophisticated that a modern mathematician might not be able to derive without the use of calculus. About 1500 years before Pythagoras, the Babylonians and Egyptians were aware of versions of the Pythagorean Theorem.

In India, their geometric history began in 900 B.C. with the publication of the Satapatha Brahmana (translated as “Brahmana of one-hundred paths”), which outlines ritual Vedic geometric constructions (historical predecessor of Hinduism) associated with the Shukla Yajurveda (third of the four canonical texts of Hinduism). These are similar to the rules for the construction of sacrificial fire altars included in the Sulba Sutras (“Aphorisms of the Chords”). This book has one of the earliest expressions of the Pythagorean Theorem despite the fact it had already been known to the Old Babylonians.

It reads:

“The diagonal rope of an oblong (rectangle) produces both which the flank and the horizontal <ropes> produce separately."

Trying to interpret this vague explanation will likely lead to throbbing headaches, but the implication of the square areas constructed on their lengths is lurking somewhere under the surface.

From there it went to China, and then to Greece, which paved the way for modern Geometry in the 17th century. Some of the more prominent figures in Greek Geometry are Thales, Pythagoras, Plato, Euclid and Archimedes. Thales was the pioneer of deduction in mathematics and wrote deductive proofs for five geometric propositions, but his proofs haven’t survived. Pythagoras is the man for whom the Pythagorean Theorem is named after, even though the knowledge of the theory predates him.

Islamic nations made several contributions to the field before it reached the age of modern Geometry. Studies of Hellenistic and Islamic texts in addition to Euclid’s Elements of Geometry produced an abundance of profound new theorems and concepts. Rene Descartes and Pierre de Fermat created analytical Geometry in the early 17th century. Girard Desargues performed a systematic study of projective Geometry during this time as well. Isaac Newton and Gottfried Wilhelm von Leibniz developed calculus. Though it wasn’t a branch of Geometry, it could be applied to it.

Although the history of a branch of mathematics might be dry, its complexity shows how much thought the founders of Geometry put into their craft. The depth of this topic is likely the reason many people can’t wrap their heads around it. If you’re caught in a bind, and have a Geometry problem you can’t solve, look to Geometry software. Perhaps the will of the greatest mathematicians will lead you to a solution and out of the woods of confusion.